Skolem Functions in Linguistics
Yoad Winter
Technion/Utrecht University
Utrecht, 5 July 2008
Syntax, Semantics, and Discourse: the Theory of the InterfaceWorkshop in memory of Tanya Reinhart
Bertnard Russell
(1872-1970)
Indefinites: existential quantifiers?
“The great majority of logicians who have dealt with this question were misled by grammar.”(Russell 1919)
My understanding: “indefinite descriptions
may behave as if they were ’referential’like proper names, but let syntax not
confuse us gentlemen – their meaning is that of existential quantifiers”.
David Hilbert
(1862-1943)
What’s wrong about existential
quantification?
Motivation: provide witness for every existential claim.
(Meyer-Viol 1995)
The Epsilon Calculus (Hilbert 1920)
∃∃∃∃x [A(x)] ⇔ A(εx.A(x))
∀∀∀∀x [A(x)] ⇔ A(εx.¬A(x))
Richard Montague (1930-1971)
Syntax as a guide for theories of meaning:
All noun phrases denote
generalized quantifiers Montague (1973)
Modern Natural Language Semantics
Russell’s distinctions – left for
philosophy of language
Hilbert’s concerns – left for
proof theory
1970s-1980s: Quantifiers Everywhere
Empirical problems for Montagovian uniformity:
Every farmer who owns a donkey beats it. (Kamp 1981, Heim 1982)
If a friend of mine from Texas had died in the fire, I would have inherited a fortune.
(Fodor and Sag 1982, Farkas 1981)
Hilbert strikes back – perhaps indefinites are (discourse) “referential” after all?
Modern Natural Language Semantics1980s-1990s: A Dynamic Turn
Early signs of SFs – branching
Henkin (1961): non-linear quantifier scope?
∀x∃z
∀y∃uΦ(x,y,z,u)
Henkin’s Semantics involves Skolem Functions (next slide).
Hintikka (1973): branching in natural language –
Some book by every author is referred to in some essay by every critic.
Branching quantifiers:
[∀x:author(x)] [∃z:book-by(z,x)]
[∀y:critic(y)] [∃u:essay-by(u,y)]referred-to-in(z,u)
(historical observation by Schlenker 2006)
What are Skolem Functions?
Functions from (tuples of) n entities to
entities.
In the logical tradition:
For example:
SF from pairs (2-tuples) over a simple domain with
elements a and b.
Skolemization (higher-order Hilbertization)
Removing existential quantifiers from formulas in Predicate Calculus.
(1) Everyone gave everyone something.
Example:
� For every two people x and y we can finda thing f(x,y) that x gave y.
The function f is an Skolem Function of arity 2 that witnesses (1).
Skolemization (cont.)
Everyone gave everyone something.
Such an R satisfies (1) and with f they satisfy (2):
(1) (2)
Suppose that R satisfies:
In linguistics: restricted quantifiers
Everyone gave everyone some present.
In the linguistic practice:
Skolem Functions are functions from n-tuples of entities and non-empty sets A to entities in A.
When n=0 (no entity arguments) the function is a choice function: it chooses a fixed element from A.
SF semantics for Hintikka’s examples?
Some book by every author is referred to in some essay by every critic.
∃f∃g [∀x:author(x)] [∀y:critic(y)]
referred-to-in(f(x,λz.book-by(z,x)), g(y,λu.essay-by(u,y)))
But the status of branching has remained undecided in the logical-linguistic literature:
- Branching generalized quantifiers (Barwise 1979, Westerstähl1987, Van Benthem 1989, Sher 1991)
- Doubts about evidence for branching (Fauconnier 1975,
Beghelli et al. 1997)
- Intermediate positions (Schlenker 2006).
(Henkin/Hintikka)More signs of SFs – functional questions
(1) Which woman does every man love?
His mother.
(2) Which woman does no man love?
His mother-in-law.
Engdahl (1980,1986), Groenendijk and Stokhof (1984), Jacobson (1999):
(1) = what is the Skolem function f such
that the following holds?
∀x [man(x) → love(x,f(x,woman))]
Tanya Reinhart (1943-2007)
“Quantification over choice
functions is a crucial linguistic
device and its precise formal
properties should be studied in
much greater depth than what I
was able to do here.”
Hilbert strikes harder: CFs(SFs) as a general semantics for indefinites and wh-elements.
Early 90s – the plot thickensReinhart (1992), early drafts of Reinhart (1997) and Kratzer (1998)
Choice functions derive the special scope properties of indefinites and wh-in-situ:
Reinhart (1992)
Reinhart’s CF thesis
If a friend of mine from Texas had died in
the fire, I would have inherited a fortune.
Reinhart’s analysis, with DRT-style closure:
Precursors semantic scope mechanisms:
Cooper (1975), Hendriks (1993)
Exceptional scope of indefinites belongs in the semantics – neither (logical) syntax nor pragmatics (Fodor and Sag) are responsible.
Summary: short history of SFs in
linguistics
– 1960s logico-philosophical foundations
1970s branching quantification
1980s functional questions
1990s – scope of indefinites, and more…
Caveat: more researchers have studied epsilon-terms
and their possible relations to anaphora, predating
current attempts – see Slater (1986), Egli (1991).
Mid 90s: new questions
□ Formalizing CFs/SFs in linguistics
□ CFs vs. general SFs
□ Empirical consequences of attributing
the scope of indefinites to semantics
□ Functional pronouns
□ General role of CFs/SFs within the DP:
definites, numerals, anaphoric
pronouns
Precise use of CFs/SFs
Empty set problem: Some fortuneteller from Utrecht arrived.
Winter (1997):
Do away with existential closure of CFs?
Kratzer (1998):
Montague-style
Hilbert / Fodor & Sag-style
CFs or general SFs?
The problem of “intermediate scope”:
(1) Every professor will rejoice if a student
of mine/his cheats on the exam.
Is there a contrast in cases like (1)?Fodor and Sag – Yes.Wide agreement nowdays – No. (Farkas, Abusch, Ruys, Reinhart, Chierchia)
Kratzer: Evidence for “referential” general SFs
Reinhart: Evidence for intermediate existential closure
Chierchia: Evidence for both
Winter (2001) – uses general SFs to block undesired effects with CFs.
Every child loves a woman he knows.
CFs or general SFs? (cont.)
Rather – the arity of the SK matches the number of bound variables within the indefinite’s restriction:
– SK0 = CF– SK1
– SK2
a woman a woman he knows
a woman who told it to him
Advantages of “semantic scope”
Ruys’ problem of numeral indefinites:
(1) If three workers in our staff have a
baby soon we will have to face hard
organizational problems.
Double scope:
1- Existential scope – island insensitive
2- Distribution scope – island sensitive
Explained by CF semantic strategy.
Winter (1997)
On-going work on SFs in Linguistics
• Indefinites/functional readings (Winter 2004)
• Branching and indefinites(Schlenker 2006)
• Donkey anaphora and SFsPeregrin and von Heusinger 2004Elbourne 2005 � Brennan 2008
Indefinites and Quantification – pictures
Bertnard Russell (1872-1970)
David Hilbert (1862-1943)
Richard Montague (1930-1971)
Tanya Reinhart (1943-2007)
indefinites
other NPs
∃∃∃∃
R/Q
ε
R
SFQ
∃∃∃∃
Qdynamic
Q
Hans Kamp
Irene Heim
References